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On the spectrum of the Steklov problem in a domain with a peak. (English. Russian original) Zbl 1171.35086
Vestn. St. Petersbg. Univ., Math. 41, No. 1, 45-52 (2008); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2008, No. 1, 56-65 (2008).
Consider the spectral Steklov problem $-\Delta_xu(x)=0\quad (x\in \Omega ),\quad \partial_N u(x)=\lambda u(x)\quad (x\in\partial \Omega\setminus \{\mathcal O\}), \tag{1}$ where $$\Omega$$ is a domain in $$\mathbb R^n$$, $$n\geq 2$$ , with a compact boundary $$\partial \Omega$$; suppose $$\partial \Omega$$ is smooth everywhere except the origin $$\mathcal O$$ of the Cartesian coordinate system $$x=(y,z)\in\mathbb R^{n-1}\times\mathbb R$$. In a neighborhood of point $$\mathcal O$$, domain $$\Omega$$ has a peak and is defined by the relations $$z>0$$, $$z^{-1-\gamma}y\in \omega$$, where $$\gamma >0$$ is the cusp factor and $$\omega$$ is a domain in $$\mathbb R^{n-1}$$ with a smooth boundary $$\partial\omega$$; $$\Delta_x$$ is the Laplace operator and $$\partial_N$$ is the derivative along the outer normal.
In the paper the problem (1) is examined in terms of the Hilbert space $$\mathcal H$$; this space is introduced as a subspace of the Sobolev space $$H^1(\Omega)$$ equipped with the norm $$\| u;\mathcal H\|=(\|\nabla_xu;L_2(\Omega)\|^2+\|u;L_2(\partial\Omega)\|^2)^{1/2}$$.
Put $$\lambda_\#=(n-3/2)^2$$mes$$_{n-1}(\omega)/$$mes$$_{n-2}(\partial\omega)$$. The main result of this paper is the following.
Theorem. The spectrum of problem (1) is located on $$[0,+\infty)$$. 1) $$\forall \gamma\in(0,1)$$ the spectrum is discrete and the eigenvalues form an infinitely increasing sequence $$0=\lambda_1\leq\lambda_2\leq\dots\leq\lambda_k\leq\dots\longrightarrow +\infty$$. 2) If $$\gamma=1$$, then the spectrum is discrete on $$[0,\lambda_\#)$$; the ray $$[\lambda_\#,+\infty)$$ constitutes the continuous spectrum. 3) If $$\gamma >1$$, then point $$\lambda =0$$ belongs to the continuous spectrum of problem (1).

##### MSC:
 35P05 General topics in linear spectral theory for PDEs
##### Keywords:
spectral Steklov problem; spectrum; peak on the boundary
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##### References:
 [1] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer, Berlin, 1985). · Zbl 0169.00206 [2] V. G. Maz’ya, Sobolev Spaces (Leningrad. Gos. Univ., Leningrad, 1984; Springer, Berlin, 1985). [3] E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen (B.G. Teubner, Stuttgart, 1983), Vol. 1, 10 Aufl. [4] M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningrad Gos. Univ., Leningrad, 1980; Reidel, Dordrecht, 1987).
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